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Let $\Delta$ be a Compact Hausdorff space in $\mathbb{C^n}$. Let $A$ be a closed sub algebra of $C(\Delta)$(space of all complex valued continuous functions on $\Delta$) which contains the constant functions. Let $\mathbb{D}$ be the open unit ball in the complex plane.

Let $A{(\mathbb{D})}=\{f\in C(\Delta): f(\Delta)\subset\mathbb{D} \}$ and $A({\bar{\mathbb{D}}})=\{f\in C(\Delta): f(\Delta)\subset\bar{\mathbb{D}} \}$.

Now let $a,b\in\Delta$, and consider the supremum $\sup_f\left\{\left|\frac{f(w_1)-f(w_2)}{1-\overline{f(w_2)}f(w_1)}\right|: f\in A{(\mathbb{D})} \right\}$. It is said that the above defined supremum is the same if we take $A({\bar{\mathbb{D}}})$ or $A{(\mathbb{D})}$. Can anyone tell how?

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    $\begingroup$ "It is said" where or by whom? $\endgroup$ – gmvh Oct 15 at 14:53
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    $\begingroup$ Line under equation (6) on page 812 of this article worldscientific.com/doi/10.1142/S0129167X97000408 $\endgroup$ – Ma18 Oct 15 at 15:30
  • $\begingroup$ Given $f\in A(\overline{\mathbb D})$ you have $rf\in A(\mathbb D)$ for every $r<1$. Then take limits $r\to 1$. $\endgroup$ – Jochen Wengenroth Oct 16 at 11:08

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